Rationality is a tool. Let's see how many things we can apply it to.

Tag Archives: math

I know this is a few days late, but I think it’s nice to have a place where all the posts are in the same place. I also really wanted to have a place to put this beautiful word cloud I made on wordle. It has all the words from all my blogathon posts, scaled to reflect the frequency of their use. I love that I seem to talk about people a lot. The other top words are pretty broad: think, just, know, like, good. They’re my go-to verbs, adjectives and adverbs. But there’s also: math, religious, questions, atheists and argue, and that all seems to describe me pretty well.

On Challenging Religious Beliefs: On why I’m working on not seeing challenging religious beliefs in social settings or online as so cringe-inducing, and why I’m glad people actually do it. (Big honking caveat: All normal social conventions like appropriateness and respect obviously need to apply)

Maaaaaaaath: How and why math is so freaking great. Includes crocheted hyperbolic spaces and some light cursing.

Safe Spaces for Racists: On what a space where people could ask “politically incorrect” questions without hurting people might look like. Note: title is meant to be catchy/provocative, not an accurate description of what I’m hoping for. By the way, if you like that post, you might like this one, called, “You Want a Space for Political Incorrectness? You Got It“, in which I announce I’m actually trying to create this space.

Brain Crack: A bunch of silly random ideas I’ve had floating around, like getting kids to teach their own classes and having churches serve as homeless shelters.

Have I scared you yet? Talking about math seems like one of the easiest ways to terrify people, make them feel stupid, and cause them actual pain. I think that’s a shame, because math is AWESOME. I’m going to try to convince you of that in the next few hundred words.

What is math?

Math is the study of patterns and logic. Any repeating pattern or system analyzed rigorously and logically can be math. The coastline of Britain? Sure! The spread of diseases? Absolutely! What the relationship between the number of sides of a shape where all sides are the same length and the area of the shape? Definitely.

Since any pattern or system is up for grabs, math is incredibly creative. You get to just pick whatever rules or approach or framework you think might yield useful or interesting results and see where they lead you. Let’s say you were interested in triangles. You could take the points of the triangles as coordinates (like (2,5) and (3,4) and stuff) and do all kinds of calculations to see what the area was. OR, you could not care at all where the points are and just take lengths of sides. OR, you could not bother with calculations or algebra at all and do the whole thing geometrically. You can even prove things with gifs!

Ok, why are you getting so excited right now?

Because that makes math AWESOME. Anything can be explored. Look, pick some rules you’ve decided to follow. See where they lead. BOOM, you’re in an entire world of your own making. Those rules are axioms. Seeing where they lead, logically, means you prove things with them, sometimes you call those conclusions theorems. And now you have a totally new mathematical world. It might as well be writing fiction or LARPing.

You think I’m kidding, but I’m not. There are 5 axioms called the Euclidean axioms. They are as follows.

A straight line can be drawn between any two points

A finite line can be extended infinitely in both directions

A circle can be drawn with any center and any radius

All right angles are equal to each other

Given a line and a point not on the line, only one line can be drawn through the point parallel to the line.

You can have all kinds of fun with just these. Take a piece of paper and see if you can convince yourself, even informally, that these seem to be true. Use crayons, markers, pencil, whatever.

Now throw them all out. Fuck ’em. We’re going to start our own mathematics with blackjack and hookers. What if instead of a straight line being the shortest distance between two points, a semicircle is. Seriously. You just invented geometry on a convex (curving inward, like the inside of a beachball) plane. It’s called hyperbolic geometry.

It looks like this.

And this.

And people freaking crochet hyperbolic curves.

You could have done basically the same thing by saying, ok, I learned in like 7th grade that all triangles have 180 degrees. Well, what if they don’t? A triangle is a shape with three sides, right? What if I want more than 180 degrees? Well, you can have whatever you want. In math, the only rule is that you have to follow your own rules. What those are, you get to decide. So draw a triangle with more than 180 degrees. Ok, it’s hard, I grant you. It seems like it would have to have more than three sides. What are we missing? What assumptions are we making? Oh! That the sides have to be straight! What if they curved out! Like a triangle you blew air into?

Congratulations, you just invented elliptical geometry. It’s the geometry that explains why planes fly like the curved line instead of the straight one:

Because the earth is curved, not flat, so the geometry changes. And triangles, just like you wanted, have more than 180 degrees.

Then Why Does Math Feel So Awful to Learn?

There are a few reasons why people hate math. For one, no one teaches math as something fun and creative. They teach it as something boring and rote, where the rules are set up beforehand and totally unchanging. To get fun math, you have to go to youtube to see people like Vi Hart make math the beautiful thing it is.

Second, because math is so abstract, it can be hard to visualize, and it makes it feel mysterious, even after you understand the problem. Like in biology, once you understand why evolution works, you get it. You know how it works. Sometimes, I’ll prove something for a class, and I’ll know it’s right, and that everything follows logically, and still not really know what I just did. For instance, visualize a line for me. That’s one dimension. What’s the two dimensional form of that? Right, a square. And three dimensions? A cube, great. And next? That question is the intellectual equivalent of moving both index fingers together in front of someone’s face, asking them to follow the fingers with their eyes, and then suddenly moving them in different directions. You just don’t know what happened to you. (For readers of Flatland, there’s a reason the sphere gets very upset when asked if there are more than three dimensions)

That shape, by the way, is called a tesseract, and it looks like the picture below in three dimensions, even thought it’s a four dimensional thing. But what does the next one look like? At some point, visualizations run out and logic and proof must take over.

Thirdly, math has a language, and it’s not an easy one to learn. There are all the symbols, for one: numbers, logical operators, less than, more than, exponent, subscript, and on and on until you think you’ll drown in them. And then there are the rules for how they fit together. This implies that. Why again? Oh yes, because this. And that makes sense because? Oh, right. But eventually, if you follow math far enough, you develop a deep respect for mathematical notation, its minimalism, its utility, and you begin to deeply distrust anyone who says, “Math would be fun, but why are there so many symbols?” (Though of course, there’s tons of math to be done without them. You get to make the rules, remember?). But you also get to criticize notation, decide that some is better than others, and take sides on Newtonian vs Leibnizian differential notation.

Proofs Without Words

But because it is in some sense, a language, I wish it was taught like one. I wish that young children read proofs without fully understanding them, just as we are encouraged to read texts in Spanish without looking up every single word. I wish we contented ourselves with the gist of the proof, the point, so that we learned to prioritize the meaning over the form, just as we may not be able to word-for-word translations of our French teacher’s request, but we know it’s time to sit down.

What’s the Point of Math? And of this blog post?

Well, math is beautiful. And fun. And creative. And the point of this blog post was to convince you of that. But if pure practicality is important to you, know that if you set up the rules properly, in that they reflect the way the universe works, then you’re going to get empirically verifiable predictions from the logical conclusions of the rules. That’s how physicists knew there had to be a Higgs Boson long before we could even in principle find one. That’s how we started building bridges.

Math can describe with incredible accuracy how the world works. But it can do so much else besides. It is a powerful discipline, and it deserves our respect. For some of us, it has commanded our reverence.

Research2BeDone has put his finger on what I agree is the most fundamental problem facing those trying to discuss social justice issues with people who aren’t familiar with the concepts involved: large inferential distances. Inferential distances are those gaps between our knowledge and the knowledge of others that make it hard to convey ideas. The example given over at Less Wrong is:

Explaining the evidence for the theory of evolution to a physicist would be easy; even if the physicist didn’t already know about evolution, they would understand the concepts of evidence, Occam’s razor, naturalistic explanations, and the general orderly nature of the universe. Explaining the evidence for the theory of evolution to someone without a science background would be much harder. Before even mentioning the specific evidence for evolution, you would have to explain the concept of evidence, why some kinds of evidence are more valuable than others, what does and doesn’t count as evidence, and so on. This would be unlikely to work during a short conversation.

Similarly, one SJ-oriented friend might be able to convey to another SJ-oriented friend why complaining about the term “cisgender” on the basis that the term is stolen from chemistry is problematic with a single step. They don’t have to explain about the way labels can empower or how words can do harm or how derailing works or what cisprivilege is, let alone privilege in general. They can just allude to all of that shared knowledge and assume it’s understood and believed. For the mathematically minded, all the lemmas have already been shown, and from there the theorem is a one step proof.

But without being able to assume all of the information, ideas and analysis that go into the Social Justice™ system, it’s much, much harder to explain what’s going on. In fact, you can’t do it directly at all. To properly make the argument, some patient and charitable soul would have to start from the beginning, the core axioms, work through all the basic approaches and forms of analysis, arguing all the way that they are legitimate and worthwhile, then showing how they apply to the situation in question, and hoping desperately that they’re still paying attention by the end. And that’s in the best case scenario, where it doesn’t disintegrate into slurs, derailing or unproductive mud-slinging before the explanation is over. Just like in math.

It seems unfair, of course, that in order just to convince someone to stop believing harmful and incorrect things, that much work has to be done. The answer seems obvious, if you already have all of the knowledge, information and assumptions. But from the other side, it isn’t at all. In fact, it’s not rational to find it obvious. Without an explanation that starts with assumptions that are in fact shared, someone who doesn’t currently agree with our fictional Social Justice Warrior doesn’t have reason to believe what they’re being told. Just as so many creationists disbelieve science because it rests on the concept of the scientific method (which they do not accept), and mathematicians dismiss proofs that require unproven assumptions (except the unproven assumptions they like), this non-SJ-er must reject the notion that “cisgender” should be a required part of hir vocabulary. (Much like hir). Note that mathematics and creationism have somewhat different truth values. It doesn’t matter; this is still how it feels from the inside to believe some things and not others.

How do we change that belief? More specifically, “How does one go about helping everyone on either side of an inferential distance gap understand each other?”

By bridging the gap! Get rid of it entirely, by meeting the person you’re talking to where they are.The following steps provide a guideline (much of which is laid out originally here):

If you require a baseline of civility or respect for the conversation to continue, make it clear from the outset. In the spirit of “you don’t have to get it to respect it,” you can demand that arguments must be in good faith and that certain words that you feel are harmful and cruel not be used for the duration of the conversation.

Find out how far back the disagreement goes by finding the most basic assumptions you agree on. Best way to do this is just to ask: “Do you agree with this? How about this?” until you figure it out.

Start from there and make your case. Try not to use jargon or specialized language that the non-SJ-er doesn’t use without definition. Step by step, get them from their column to yours. If you find you can’t prove your point from that far back, it’s time to ask yourself again why you believe what you believe.

Since you’re taking them through a long series of steps, be willing to accept compromise. Be happy if you took them through some of the steps, even if you had to stop there. It’s all a journey.

Similarly, since going through this many steps is hard, see if there are any places to make it easier. Skip nonvital steps. Condense and simplify if you get the opportunity. This will both help your argument and teach you what parts of your argument are required for the rest to stand and what parts are not.

Being able to construct your own argument from first principles is great. Being able to construct the other side’s is even better. It allows for so much insight into why they don’t agree with you in the first place, which makes you more charitable and more effective when you’re looking to win them over.

The tips might look intimidating, but the important part has only three steps. It’s really that simple. It’s hard to be perfectly persuasive all throughout the argument, it’s hard to make an argument that extensive, and it’s frustrating to do it over and over again. But it is simple. For those willing to do it, arguing with people who have entirely different assumptions is just the task of laying out a path, slowly but surely, from one set of beliefs to another.

I do not deny for a second that it can seem like a waste of time, that it can be painful, and that rather more often than we might hope, the people we’re arguing with are not arguing in good faith. That is why we leave it to individuals to decide whether it is worth their time and effort. But those not willing to do this kind of work should not stand in its way. They should not base their arguments on assumptions others do not share and be surprised when they are not understood. They should not make it more difficult for others to do the challenging work by interrupting ongoing conversations with jeering and mockery. And most of all, while there are perfectly good reasons to stop being able to have a conversation or to not enter one in the first place, no one should engage in arguments with people who might be persuaded if they have no intention of taking the process seriously. Ideas rise and fall every day in the public sphere, and there’s no reason to lose arguments or adherents because some don’t think the work of public reason is worth doing properly.

Argledy-Bargledy. I was doing so well with blogging more frequently! But then I got busy. I fully intend to return to my series on Better Arguing as well as the several other things I have planned (including a discussion of circumcision, an exegesis on a Torah portion and a Rationalist Manifesto on guns and gun control among them), but in the meantime, here’s something I’ve been thinking about for a while.

People use the phrases “more unique” and “very unique” all the time, and the grammarians, traditionalists and precision-fetishists all hate it. Unique is a binary descriptor, they cry, denoting a singular nature, unparalleled, different than anything else. How could anything possibly be more or very unique? Those who use the phrases tend to rebut that language is what it’s users make of it and that the meddlers should butt out. But I happen to find mathematics far more interesting than ye olde prescriptivist vs descriptivist debate, so I prefer to tackle the question that way.

Let us imagine a line, like a number line, for some property of objects. Maybe its color, or size, or frequency on earth, or price or chance of being ejected from a cockpit. Points on the line correspond to a value of that property, even if the property isn’t a continuous one like size. Now let’s say we have as many lines as properties, and we can graph things by going to the point on each line where their property matches up. We get tons of points, some of which are in clusters, because they are similar in some way, like apples would congregate around an area in this n-dimensional graph that had a certain number for redness and crunchiness and edibleness.

When someone says that something’s unique, the weakest formulation of that idea is that the object or thing in question has a dot where no other dot is. But that’s boring, because my iPhone is different that everyone else’s at least slightly, but it’s not unique in a meaningful sense. So the word unique already corresponds to something that’s not really a binary, because some things are meaningfully unique (maybe “more unique”?) and others are not, even if they are both unique. Mostly, when people say that something’s unique, they mean it’s far away from the cluster where it would normally be found, usually on markers of goodness or excellence.

Of course, far away is one of those pesky continuous sort of things. Something can be more further away or less further away, if I am allowed to destroy the English language further that way. Thus we have that the further something is away from the cluster where we expect it, the more unique it is. And so “more unique” makes perfect sense after all.